Reference heavy flavour cross sections in pp collisions at √ s = 2.76 TeV, using a pQCD-driven √ s-scaling of ALICE measurements at √ s = 7 TeV R. Averbeck, 1 N. Bastid, 2 Z. Conesa del Valle, 3 P. Crochet, 2 A. Dainese, 4 and X. Zhang 2, 5 1 Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum f¨ ur Schwerionenforschung, Darmstadt, Germany 2 LPC, Clermont-Ferrand, France 3 CERN, Geneva, Switzerland 4 INFN – Sezione di Padova, Padova, Italy 5 CCNU, Wuhan, China We provide a reference in proton–proton collisions at the energy of the Pb–Pb 2010 run at the LHC, √ s = 2.76 TeV, for the p t -differential production cross section of D 0 , D + , and D *+ mesons in |y| < 0.5, of electrons from heavy flavour decays in |y| < 0.9, and of muons from heavy flavour decays in 2.5 < y < 4. The reference is obtained by applying a pQCD-driven scaling (based on the FONLL calculation) to ALICE preliminary data at √ s = 7 TeV. In order to validate the procedure, we scale the D meson cross section to √ s = 1.96 TeV and compare to the corresponding measurements from the CDF experiment. arXiv:1107.3243v2 [hep-ph] 19 Jul 2011
Transcript
Reference heavy flavour cross sections in pp collisions at√
s = 2.76 TeV, using a pQCD-driven√s-scaling of ALICE measurements at
√s = 7 TeV
R. Averbeck,1 N. Bastid,2 Z. Conesa del Valle,3 P. Crochet,2 A. Dainese,4 and X. Zhang2, 5
1Research Division and ExtreMe Matter Institute EMMI,GSI Helmholtzzentrum fur Schwerionenforschung, Darmstadt, Germany
4INFN – Sezione di Padova, Padova, Italy5CCNU, Wuhan, China
We provide a reference in proton–proton collisions at the energy of the Pb–Pb 2010 run at the LHC,√
s =2.76 TeV, for the pt-differential production cross section of D0, D+, and D∗+ mesons in |y|< 0.5, of electronsfrom heavy flavour decays in |y|< 0.9, and of muons from heavy flavour decays in 2.5 < y < 4. The referenceis obtained by applying a pQCD-driven scaling (based on the FONLL calculation) to ALICE preliminary dataat√
s = 7 TeV. In order to validate the procedure, we scale the D meson cross section to√
s = 1.96 TeV andcompare to the corresponding measurements from the CDF experiment.
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I. INTRODUCTION
In 2010, the Large Hadron Collider (LHC) delivered large samples of proton–proton (pp) collisions at√
s = 7 TeV and of Pb–Pb collisions at
√sNN = 2.76 TeV. A pp reference at
√s = 2.76 TeV is required, in order to compare heavy flavour production
in Pb–Pb and pp collisions via the nuclear modification factor of the pt distributions
RAA(pt) =1〈TAA〉
dNAA/dpt
dσpp/dpt. (1)
Here, 〈TAA〉 is the average value of the nuclear overlap function in a given Pb–Pb centrality class, NAA is the production yieldper event of the considered particle in that class, and σpp is its production cross section in pp collisions at the same energy.
In this note, we show that state-of-the-art perturbative QCD calculations provide an accurate guidance for extrapolating tolower energy the pt-differential cross sections measured at 7 TeV. It was already shown [1], using the MNR [2] NLO pQCDcalculations at the energies
√s = 5.5 and 14 TeV, that, despite the large spread for the cross section at a given energy with
different values of the heavy quark masses and factorization/renormalization scales, the ratio of the cross sections at the twoenergies is much less dependent on the choice of the calculation parameters. We now use the Fixed Order Next-to-Leading Log(FONLL) calculations [3], and apply the resulting scaling factor to preliminary ALICE cross section measurements at 7 TeV:D mesons (D0, D+, and D∗+) at mid-rapidity and single leptons (electrons at mid-rapidity and muons at forward rapidity) fromcharm and beauty hadron decays. In order to validate the procedure, we scale the D meson cross sections to
√s = 1.96 TeV and
compare to the corresponding measurements from the CDF experiment [4].After reporting the ALICE heavy flavour cross section measurements at
√s = 7 TeV (section II), we describe (section III) the
procedure adopted for the energy scaling, and (section IV) the scaling factors and resulting cross sections at 2.76 TeV.
II. ALICE HEAVY-FLAVOUR PRODUCTION MEASUREMENTS AT√
s = 7 TeV
Here we briefly present the ALICE measurements at√
s = 7 TeV that will be used for the scaling.The preliminary results on the D0, D+, and D∗+ cross sections at
√s = 7 TeV, measured at central rapidity (|y| < 0.5) using
the decay channels D0 → K−π+, D+ → K−π+ π+, D∗+ → D0 π+ → K−π+ π+, are shown in Fig. 1. The data are comparedto pQCD predictions based on the FONLL [3, 5] and GM-VFNS [6] calculations. These results are described in [7, 8]. Let usrecall that these measurements correspond to the direct charm production, as they were corrected for the B-mesons feed-downcontribution.
Fig. 2 shows the preliminary cross section of electrons from heavy flavour decays for pp collisions at√
s = 7 TeV and thecorresponding prediction from the FONLL pQCD calculation [3, 5]. A reasonable agreement between data and model calculationis observed. The analysis procedure is discussed in [10].
The inclusive pt and η differential cross section of muons from heavy flavour decays, in the rapidity range 2.5 < y < 4, in ppcollisions at
√s = 7 TeV is displayed in Fig. 3 [13], along with the corresponding FONLL prediction [3, 5].
III. ENERGY SCALING PROCEDURE
In order to scale the ALICE 7 TeV cross-sections to a given energy we consider the scaling factors provided by differenttheoretical calculations. The FONLL [3, 5] driven scaling is set as our reference scaling and is evaluated considering the differentsets of scales (factorization scale µF , renormalization scale µR) and quark masses (mc and mb). We consider the standardparameter variations that are used to evaluate the theoretical uncertainty on the charm and beauty production cross sections (seee.g. [11, 12]):
t,Q = mt,Q.The procedure to compute the FONLL scaling factor from 7 TeV to an energy of α TeV is:
1. Rebin the FONLL predictions for σ(α) and σ(7) for the different sets of scales (µF , µR), and quark masses (mc and mb)according to the ALICE 7 TeV pt binning for each observable.
2
GeV/c t
p0 2 4 6 8 10 12 14
b/G
eV
/cµ
|y|<
0.5
| t / d
pσ
d
110
1
10
210
310+π
K→ 0D
1 = 7 TeV, 1.6 nbspp,
PW
G3P
relim
inary
024
= 62.3 mbMB
σ
7% global norm. unc. (not shown)±
ALICE Preliminary
stat. unc.
syst. unc.
FONLL
GMVFNS
GeV/c t
p0 2 4 6 8 10 12 14
b/G
eV
/cµ
|y|<
0.5
| t / d
pσ
d
110
1
10
210
310+π +π
K→ +D
1 = 7 TeV, 1.6 nbspp,
PW
G3P
relim
inary
025
= 62.3 mbMB
σ
7% global norm. unc. (not shown)±
ALICE Preliminary
stat. unc.
syst. unc.
FONLL
GMVFNS
GeV/c t
p0 2 4 6 8 10 12 14
b/G
eV
/cµ
|y|<
0.5
| t / d
pσ
d
110
1
10
210
310+π +π
K→ +π 0 D→ *+D
1 = 7 TeV, 1.6 nbspp,
PW
G3P
relim
inary
026
= 62.3 mbMB
σ
7% global norm. unc. (not shown)±
ALICE Preliminary
stat. unc.
syst. unc.
FONLL
GMVFNS
FIG. 1: ALICE D0, D+, and D∗+ pt differential preliminary cross sections in |y|< 0.5 at√
s= 7 TeV [7]. The FONLL [3, 5] and GM-VFNS [6]predictions are compared to the data.
2. Estimate the FONLL σ(α)/σ(7) ratio per observable1 considering that:
• The central value is the ratio of the central predictions at both energies and
• its uncertainty is defined by the envelope (spread) of the ratio of the calculations for the different sets of parameters.Note that for a given quark flavour we can consider that the theoretical calculation parameters are correlated (equal)at different energies. However, we do not assume they are equal for charm and beauty.
1 This means that for single leptons this is the ratio of charm + bottom contributions.
3
), |
y|<
0.8
2d
y (
mb
/(G
eV
/c)
T/d
pσ
2 d
Tp
π1
/2
810
710
610
510
410
310
210
110
11
Ldt = 2.6 nb∫ = 7 TeV, spp,
ALICE Preliminary
e→ALICE b,c
e→FONLL b,c
7% normalization error
(GeV/c)T
p
0 2 4 6 8 10
Data
/FO
NL
L
0
0.5
1
1.5
2
FIG. 2: ALICE heavy flavour decay electron pt differential production cross section for pp collisions at√
s = 7 TeV [10]. The FONLL pQCDcalculation [3, 5] is compared to the data.
FIG. 3: ALICE pt and η differential production cross section of muons from heavy flavour decays, in 2.5 < y < 4, in pp collisions at√
s = 7TeV (symbols) [13]. The results are compared to FONLL predictions [3, 5].
3. Multiply the ALICE 7 TeV cross-sections by the FONLL σ(α)/σ(7) binned ratio.
4. Propagate the uncertainties:
4
• on the FONLL ratios,
• on the uncertainties of the 7 TeV measurement,
• combine these uncertainties.
The considered cross-checks of the scaling procedure are :
1. Interpolate to Tevatron energy (pp at√
s = 1.96 TeV) to compare to the D meson measurements by the CDF Collabora-tion [4];
2. Compare the scaling factor from FONLL to that obtained from other (Fixed Order) pQCD calculations (NLO MNR [2],GM-VFNS [6]).
IV. RESULTS
A. D mesons
1. Scaling factor to√
s = 2.76 TeV
The FONLL scaling factors for D0, D+, and D∗+were calculated as described in the previous section and are shown in Figs. 4, 5,and 6 (left-hand panels) together with their respective relative uncertainties (right-hand panels). The scaling factor obtained withthe different sets of scales are drawn with solid lines, while the resulting global scaling is depicted by a yellow filled band.The central value of the scaling is obtained with µF/µ0 = µR/µ0 = 1. The values of the scales for the other sets are reported inthe legend (µF/µ0, µR/µ0). We can observe that the scaling factor depends mainly on the value of the factorization scale, withalmost no dependence on the renormalization scale. This is due to the fact that, for the same heavy quark pt, different Bjorkenx ranges are probed at 2.76 and at 7 TeV, and changing the factorization scale affects the x dependence of the parton distributionfunctions (PDFs). The scaling factor does not depend on the value used for the charm quark mass in the calculation, as shown inFig. 7 (right) using the MNR NLO calculation. The scaling has a large pt dependence in the low pt region, where it varies from afactor of ≈ 0.8 at pt ≈ 1 GeV/c to ≈ 0.4 at pt ≈ 5 GeV/c, while at higher pt the variation less pronounced. The average scalingfactor calculated for the D0, D+, and D∗+mesons is very similar, while some small variations can be observed on the uncertaintybands.
2. Influence of the theoretical calculation: MNR and GM-VFNS vs FONLL
The FONLL direct charm scaling has been tested by comparing the calculation to the one obtained with the MNR [2] andGM-VFNS [6] calculations.
a. MNR calculation: The MNR scaling factor for the different sets of scales is shown in Fig. 7 (left). Fig. 7 (right) showsthat the influence of varying the charm quark mass from 1.3 to 1.7 is negligible. The comparison of the MNR and FONLLcalculations for D0 mesons (Fig. 8) demonstrates that, as expected, the scaling factors agree with each other, and that theuncertainties are larger for the MNR case. Therefore, from now on we will drop the MNR case for this exercise.
b. GM-VFNS calculation: We obtain the GM-VFNS scaling factor considering that the three calculation parameters (therenormalization scale, the factorization scale for initial state singularities and the factorization scale for final state singularities)do not depend on the value of
√s, as for the FONLL case, with the difference that the latter considers only the factorization and
renormalization scales. The D0 meson scaling to 2.76 TeV is shown in Fig. 9, where the calculation parameters are varied within1/2 (h), 1 and 2 times the standard parameters. The spread of the ratio evaluated for the different parameters indicate the scalinguncertainties.
The comparison of the D0 FONLL and GM-VFNS scalings is shown in Fig. 10 for different pt binnings. The agreement ofthe scaling central values and their uncertainties for the considered pt bins is striking. We can then conclude that there is noneed to do all the scalings both with GM-VFNS and FONLL calculations since their energy evolution (and uncertainties) are inagreement.
3. Comparison to CDF measurements in pp at√
s = 1.96 TeV
In this section, we show the comparison of the CDF [4] and the ALICE measurements scaled to 1.96 TeV.We evaluated the scaling factor from 7 TeV to 1.96 TeV with the FONLL calculations. These estimates were used to scale the
D0, D+, and D∗+ cross sections measured by ALICE to 1.96 TeV. Figures 11, 12, and 13 present the comparison of these scalings
5
[GeV/c] t
p0 2 4 6 8 10 12
( 7
TeV
)i
σ (
2.7
6 T
eV
) /
iσ
0
0.2
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0.8
1
1.2
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0µ/
Rµ
0µ/
Fµ
1 1 0.5 0.5
2 2 2 1
1 2 1 0.5 0.5 1
scaling (2.76 TeV / 7 TeV)0
D
[GeV/c] t
p0 2 4 6 8 10 12
Re
lati
ve
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ce
rta
inty
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scaling (2.76 TeV / 7 TeV)0
D
FIG. 4: D0 FONLL scaling to 2.76 TeV from 7 TeV (7 TeV data binning).
[GeV/c] t
p0 2 4 6 8 10 12
( 7
TeV
)i
σ (
2.7
6 T
eV
) /
iσ
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0µ/
Rµ
0µ/
Fµ
1 1
0.5 0.5
2 2
2 1
1 2
1 0.5
0.5 1
scaling (2.76 TeV / 7 TeV)+
D
[GeV/c] t
p0 2 4 6 8 10 12
Re
lati
ve
Un
ce
rta
inty
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
scaling (2.76 TeV / 7 TeV)+
D
FIG. 5: D+ FONLL scaling to 2.76 TeV from 7 TeV (7 TeV data binning).
[GeV/c] t
p0 2 4 6 8 10 12
( 7
TeV
)i
σ (
2.7
6 T
eV
) /
iσ
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0µ/
Rµ
0µ/
Fµ
1 1
0.5 0.5
2 2 2 1
1 2
1 0.5
0.5 1
scaling (2.76 TeV / 7 TeV)+
D*
[GeV/c]t
p0 2 4 6 8 10 12
Re
lati
ve
Un
ce
rta
inty
1
0.8
0.6
0.4
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0
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0.4
0.6
0.8
1
scaling (2.76 TeV / 7 TeV)+
D*
FIG. 6: D∗+ FONLL scaling to 2.76 TeV from 7 TeV (7 TeV data binning).
and the CDF measurements2. The ratio of the CDF / ALICE points is also shown (right-hand panels). The yellow (orange) bands
2 Here the CDF cross sections have been rebinned to match the ALICE 7 TeV preliminary measurements binning.
6
[GeV]T
D meson p0 5 10 15 20 25 30
=7 T
eV
) in
|y|<
1s
(T
/dp
σ=
2.7
6 T
eV
) / d
s(
T/d
pσ
d 0
0.2
0.4
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1
1.2
1.40
µ/R
µ, 0
µ/F
µ, cm
1.5, .5, .5
1.5, .5, 1
1.5, 1, .5
1.5, 1, 1
1.5, 1, 2
1.5, 2, 1
1.5, 2, 2
[GeV]T
D meson p0 5 10 15 20 25 30
=7
Te
V)
in |
y|<
1s
(T
/dp
σ=
2.7
6 T
eV
) /
ds
(T
/dp
σd
0
0.2
0.4
0.6
0.8
1
1.2
1.40
µ/R
µ, 0
µ/F
µ, cm
1.3, 1, 1
1.5, 1, 1
1.7, 1, 1
FIG. 7: D0 MNR scaling to 2.76 TeV from 7 TeV (fine binning): scales dependence (left) and charm quark mass dependence (right).
[GeV/c]t
D meson p0 5 10 15 20 25 30
Ra
tio
2
.76
Te
V /
7 T
eV
0
0.2
0.4
0.6
0.8
1
1.2
1.4
[GeV/c]t
D meson p0 5 10 15 20 25 30
Ra
tio
2
.76
Te
V /
7 T
eV
0
0.2
0.4
0.6
0.8
1
1.2
1.4MNR
FONLL
FIG. 8: D0 MNR scaling to 2.76 TeV from 7 TeV (yellow band) compared to the FONLL scaling (purple hatched band).
describe the maximum (conservative) uncertainty on this ratio, considering that the CDF and ALICE scaled uncertainties areuncorrelated (correlated), i.e. considering the ratios of the upper-CDF to lower-ALICE (upper-CDF to upper-ALICE) and viceversa. Overall, these ratios are compatible with unity (within somewhat large uncertainties), demonstrating that the scalingprocedure is reliable. We note that, for the D∗+ case, although compatible with 1 within 1.2 sigma, the ratio is centred at 1.5:rather than to an anomaly of the scaling, which is practically the same for all D mesons, this could related the observation thatratio D0/D∗+ measured by ALICE at 7 TeV is larger than that measured by CDF at 1.96 TeV [8].
4. Results at 2.76 TeV and relative uncertainties
Finally, we can scale the ALICE D0, D+ and D∗+ measurements at 7 TeV to 2.76 TeV considering the FONLL scaling factorsevaluated in section IV A 1. The scaled cross sections are presented in Fig. 14 for D0 (top-left), D+ (top-right) and D∗+ (bottom).
7
[GeV/c]t
p5 10 15 20 25 30
(7 T
eV
)i
σ (
2.7
6 T
eV
) /
iσ
0
0.1
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0.4
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1111 (central)
112
11h
121
122
1h1
1hh
211
212
221
222
h11
h1h
hh1
hhh
scaling (2.76 TeV / 7 TeV)0
D
FIG. 9: D0 GM-VFNS scaling to 2.76 TeV from 7 TeV considering that the scales are correlated vs energy. The yellow band representsthe global scaling and its uncertainty. The values of the renormalization scale, the factorization scale for initial state singularities and thefactorization scale for final state singularities are reported in the legend.
[ GeV/c]t
p5 10 15 20 25 30
(7 T
eV
)i
σ (
2.7
6 T
eV
) /
iσ
0
0.1
0.2
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0.5
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0.9
1
FONLL
GMVFNS
scaling0D
[GeV/c]t
p3 4 5 6 7 8 9 10 11 12
(7 T
eV
)i
σ (
2.7
6 T
eV
) /
iσ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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1
FONLL
GMVFNS
scaling0
D
FIG. 10: Comparison of D0 GM-VFNS and FONLL scaling to 2.76 TeV from 7 TeV with fine (left) and 7 TeV preliminary cross section binning(right).
The influence of the 7 TeV data systematics and of the FONLL interpolation systematics on the global scaled systematics is alsodepicted, showing the relatively small contribution of the FONLL scaling uncertainties.
B. Heavy flavour decay electrons in |y|< 0.8
The FONLL scaling factor from√
s = 7 TeV to√
s = 2.76 TeV is calculated for electrons from charm and beauty decaysusing the approach described in section III. For all charm related calculations shown in this section we assume that neutral Dmesons contribute 70% to the total electron yield and the remaining 30% originate from charged D meson decays.
In the case of electrons from heavy flavour decays an additional complication arises from the fact that the relative contributionsfrom charm and beauty decays change as function of pt and are not known a priori. Therefore, it is crucial to compare the scalingfor electrons from charm decays and beauty decays separately before evaluating a combined scaling function. This comparisonis shown in the left panel of Fig. 15 for the default choices of quark masses, parton distribution function, and scales µR andµF . The scaling factors for electrons from charm and beauty decays are almost the same except for the region of low transverse
8
[GeV/c]t
p0 2 4 6 8 10 12 14 16 18 20
b / G
eV
/c]
µ [ t
/ d
pσ
d
210
110
1
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210CDF data (PRL91), rebinned
ALICE Preliminary 7 TeV
scaled to 1.96 TeV
Total syst. unc.
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meson0
D
[GeV/c]T
D meson p4 5 6 7 8 9 10 11 12
CD
F m
easu
rem
en
t / A
LIC
E F
ON
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ex
trap
ola
tio
n
0
0.2
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1
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1.6
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2
FIG. 11: Left: Comparison of D0 ALICE 7 TeV measurements scaled to 1.96 TeV with the CDF measurements. Right: ratio of thesetwo. The yellow (orange) band describes the maximum (conservative) uncertainty on the ratio, considering that the CDF and ALICE scaleduncertainties are uncorrelated (correlated). Note that the first CDF data point, 5.5 < pt < 6 GeV/c, is compared to the ALICE data point for5 < pt < 6 GeV/c.
[GeV/c]t
p0 2 4 6 8 10 12 14 16 18 20
b /
Ge
V/c
]µ
[ t /
d p
σ d
210
110
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210CDF data (PRL91), rebinned
ALICE Preliminary 7 TeV
scaled to 1.96 TeV
Total syst. unc.
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meson+
D
[GeV/c]T
D meson p5 6 7 8 9 10 11 12
CD
F m
easu
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LIC
E F
ON
LL
ex
trap
ola
tio
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FIG. 12: Same as in Fig. 11, for D+.
[GeV/c]t
p0 2 4 6 8 10 12 14 16 18 20
b / G
eV
/c]
µ [ t
/ d
pσ
d
210
110
1
10
210CDF data (PRL91), rebinned
ALICE Preliminary 7 TeV
scaled to 1.96 TeV
Total syst. unc.
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D* meson
[GeV/c]T
D meson p0 2 4 6 8 10 12
CD
F m
easu
rem
en
t / A
LIC
E F
ON
LL
ex
trap
ola
tio
n
0
0.5
1
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2
2.5
3
FIG. 13: Same as in Fig. 11, for D∗+.
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p0 2 4 6 8 10 12 14
b / G
eV
/c]
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/ d
pσ
d
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scaled to 2.76 TeV
Total syst. unc.
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meson0D
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b / G
eV
/c]
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/ d
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scaled to 2.76 TeV
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meson+D
[GeV/c] t
p0 2 4 6 8 10 12 14
b / G
eV
/c]
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/ d
pσ
d
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210 ALICE Preliminary 7 TeV
scaled to 2.76 TeV
Total syst. unc.
ALICE syst. unc.
FONLL scaling unc.
meson+D*
FIG. 14: D0 (top-left), D+ (top-right), and D∗+ (bottom) ALICE 7 TeV measurement scaling to 2.76 TeV. The purple points and boxesrepresent the interpolated cross section and its global systematics. The blue empty (magenta filled) boxes indicate the 7 TeV measurement(FONLL interpolation) contribution to the systematics.
momenta (pt < 2 GeV/c), where the relative contribution from beauty decays to the total heavy flavour decay electron yield istiny.
The agreement of the charm and beauty decay electron scaling factors as observed in FONLL justifies to calculate onecombined scaling factor for heavy flavour decay electrons as the ratio of the sum of charm and beauty decay electron crosssections at 2.76 TeV relative to the sum at 7 TeV. This combined scaling factor is shown in the right panel of Fig. 15 for thedefault FONLL parameters.
In the following, we discuss the uncertainties of the combined heavy flavour decay electron scaling factor due to the uncer-tainties of the FONLL parameters, i.e. the uncertainties of the quark masses and the scale parameters µR and µF .
The dependence of the scaling factor on the quark masses turns out to be negligible as demonstrated in Fig. 16. Here, weconsider quark masses of mc = 1.5±0.2 GeV and mb = 4.75±0.25 GeV.
As for D mesons, the dependence of the heavy flavour decay electron scaling factor on the choice of the FONLL scaleparameters is addressed in Figs. 17 and 18. In addition to the FONLL default scales µR = µF = µ0, we calculate the scalingfactor for the scale values 0.5µ0 and 2µ0.
First, we assume that the scales are the same for charm and beauty. The resulting heavy flavour electron scaling factors areshown in Fig. 17. The spread of the calculations with different scaling factors is of order 10% or less for an electron pt largerthan 2 GeV/c.
However, it can not be excluded that the scale parameters vary independently for charm and beauty. To quantify the uncertaintydue to that possibility we show in Fig. 18 the heavy flavour electron scaling factors one can calculate with individual choices forthe scale parameters for charm and beauty in FONLL. The resulting spread of these calculations is within the envelope of thecalculations for which the same scale parameters have been used for charm and beauty.
Figure 19 shows the cross section of electrons from heavy flavour decays measured by ALICE at√
s = 7 TeV and the resultof the scaling to 2.76 TeV.
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FIG. 15: FONLL scaling factor from 7 TeV to 2.76 TeV for electrons from charm and beauty decays, respectively (left panel). CombinedFONLL scaling factor for electron from heavy flavour decays (right panel).
C. Heavy flavour decay muons in 2.5 < y < 4
The FONLL scaling factor that will be applied to the pt-differential cross section of muons from heavy flavour decay measuredin pp collisions at
√s = 7 TeV, in order to obtain the reference cross section at
√s = 2.76 TeV, was determined according to the
procedure described in section III. The only difference with respect to the case of electrons (section IV B) is that two differentrapidity ranges are used for the FONLL calculations: |y| < 0.8 for electrons and 2.5 < y < 4 for muons. Figure 20 shows thescaling factor obtained by combining different sets of c and b quark masses and assuming that quark masses are unchanged at2.76 TeV and 7 TeV. The scaling factor depends strongly on pt, in particular in the low pt range (pt < 2 GeV/c). It decreasesfrom about 0.5 to 0.2 in the pt range 0–6 GeV/c and tends to saturate (Fig. 20, left panel). The relative scaling factor, which givesthe relative uncertainty, is depicted in the right panel of Fig. 20. Changes in the quark masses introduce a systematic uncertaintyless than 5% for pt < 2 GeV/c, which can be neglected at higher pt (range of interest for the measurement of the cross sectionof muons from heavy flavour decay).
As in the other cases, it was assumed that the pQCD scales do not change with√
s. As for electrons at central rapidity, theinfluence of the pQCD scales variation on the FONLL scaling factor was investigated in two cases: a) same scales for charmand beauty (correlated scales, colour lines in Fig. 21); b) different scales for charm and beauty (uncorrelated scales, blacklines in Fig. 21). Very similar results are obtained with correlated or uncorrelated scales for charm and beauty. At low muonpt (< 2 GeV/c) the uncertainty on the scaling factor reaches about 40%, while in the pt > 2 GeV/c range it is below 10%,independently of pt.
In summary, the FONLL scaling factor as a function of pt obtained for different sets of quark masses (blue boxes) and pQCDscales (red boxes), as just discussed, is shown in Fig. 22 (left panel). The relative scaling factor is also shown in the right panelof the figure. For the systematic uncertainty from energy scaling, we consider the spread of the ratio obtained with the differentsets of parameters (yellow band).
Figure 23 shows the cross section of muons from heavy flavour decays obtained by scaling to√
s = 2.76 TeV the ALICEmeasurement at 7 TeV shown in Fig. 3.
V. CONCLUSIONS
We have presented a procedure to define the√
s-scaling factors for heavy flavour production cross sections in pp collisions atLHC energies. The scaling is based on perturbative QCD calculations, as implemented in the FONLL scheme, which describedreasonably well heavy flavour production as measured at the Tevatron and at the LHC. For D mesons and heavy flavour decayleptons, the scaling uncertainty from 7 to 2.76 TeV is of about 40% for pt < 2 GeV/c and < 1 GeV/c respectively, and it decreasestowards larger momenta, reaching a level below 10%. For D mesons, the scaling was verified by comparing the scaled ALICE7 TeV measurement to data by the CDF experiment at 1.96 TeV.
By applying the scaling to ALICE preliminary measurements at 7 TeV for D0, D+, D∗+, electrons and muons from heavyflavour decay, we have provided reference cross sections in pp collisions at 2.76 TeV.
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FIG. 16: FONLL scaling factor from 7 TeV to 2.76 TeV for electrons from heavy flavour decays with different values for the bare quark masses(left panel). Variation of scaling factors obtained with different quark masses relative to the scaling factor calculated with the default quarkmasses (right panel).
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/2T
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T, 2m
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scales f, r = m
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T, m
Tscales f, r = 2m
T, 2m
Tscales f, r = m
/2T
, mT
scales f, r = m
T/2, m
Tscales f, r = m
FIG. 17: FONLL scaling factor from 7 TeV to 2.76 TeV for electrons from heavy flavour decays with different values for the scale parametersµR and µF , which are chosen to be the same for charm and beauty (left panel). Variation of scaling factors obtained with different scaleparameters relative to the scaling factor calculated with the default scale parameters (right panel).
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FIG. 18: FONLL scaling factor from 7 TeV to 2.76 TeV for electrons from heavy flavour decays with different values for the scale parametersµR and µF , where the scale parameters are allowed to vary independently for charm and beauty (left panel). Variation of scaling factorsobtained with different scale parameters relative to the scaling factor calculated with the default scale parameters (right panel).
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], |
y|<
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mb
/(G
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π1/2
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= 7 TeV scaled to 2.76 TeVsALICE Preliminary data at
= 7 TeVs e →ALICE b,c
e scaled to 2.76 TeV→b,c
FIG. 19: Cross section of electrons from heavy flavour decays measured by ALICE at√
s= 7 TeV [10] and the result of the scaling to 2.76 TeV.
FIG. 20: Left: FONLL scaling factor from 7 TeV to 2.76 TeV for the measurement of pt-differential cross section of muons from heavy flavourdecay with different combinations of quark masses indicated on the figure; right: corresponding relative systematic uncertainty.
Acknowledgments
The authors would like to thank M. Cacciari for fruitful and stimulating discussions on the scaling procedure, as well asM. Cacciari and H. Spiesberger for providing the numerical cross sections from the FONLL and GM-VFNS calculations.
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FIG. 21: Left: FONLL scaling factor from 7 TeV to 2.76 TeV for the measurement of pt differential cross section of muons from heavy flavourdecay with different combinations of QCD scales as indicated on the figure; right: corresponding relative systematic uncertainty. See the textfor more detail.
FIG. 22: Left: FONLL scaling factor from 7 TeV to 2.76 TeV for the measurement of pt differential cross section of muons from heavyflavour decay with different combinations of QCD scales (red boxes) and quark masses (blue boxes). The yellow band is the total systematicuncertainty; right: corresponding relative systematic uncertainty.
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